Commutation
Reorder operands in conjunctions and disjunctions without changing meaning
P∧Q ⊢ Q∧PCommutation is one of the most intuitive logical rules, reflecting how the order of terms doesn't matter for certain operations. Just like addition and multiplication in arithmetic (3+5 = 5+3), logical AND and OR operations are commutative: the order of the operands can be swapped without changing the truth value.
Two Commutative Laws:
• P∧Q ≡ Q∧P - Conjunction is commutative • P∨Q ≡ Q∨P - Disjunction is commutative
Why It Works:
Both AND and OR operations depend only on the truth values of their operands, not their order. 'P and Q' means exactly the same as 'Q and P', and 'P or Q' means exactly the same as 'Q or P'.
P ∧ Q ≡ Q ∧ P and P ∨ Q ≡ Q ∨ PConjunction Commutation
Original:
Commuted:
Disjunction Commutation
Original:
Commuted:
Complex Example
Complex:
Commuted:
Key Point: Commutation shows that the order of operands does not affect the truth value in conjunction and disjunction.
Example 1: Weather Conditions(Symmetric observations)
Original order:
Commuted order:
Explanation: Both statements describe the same weather conditions - the order of observation does not change the meaning.
Example 2: Academic Requirements(Course prerequisites)
Requirements A:
Requirements B:
Explanation: Academic flexibility allows either course order - commutation shows that both phrasings represent the same graduation requirement.
Example 3: System Dependencies(Technical requirements)
Setup option 1:
Setup option 2:
Explanation: When system components are independent, commutation shows that installation order flexibility without affecting the final configuration.
Important: Not all logical operations are commutative. Here are key exceptions:
❌ Implication (→)
P→Q ≢ Q→P"If it rains, then it's wet" ≠ "If it's wet, then it rains"
❌ Subtraction-like Operations
P∧¬Q ≢ ¬Q∧P(different emphasis)While logically equivalent, the emphasis may differ